Problem: Is ${415431}$ divisible by $9$ ?
Explanation: A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {415431}= &&{4}\cdot100000+ \\&&{1}\cdot10000+ \\&&{5}\cdot1000+ \\&&{4}\cdot100+ \\&&{3}\cdot10+ \\&&{1}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {415431}= &&{4}(99999+1)+ \\&&{1}(9999+1)+ \\&&{5}(999+1)+ \\&&{4}(99+1)+ \\&&{3}(9+1)+ \\&&{1} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {415431}= &&\gray{4\cdot99999}+ \\&&\gray{1\cdot9999}+ \\&&\gray{5\cdot999}+ \\&&\gray{4\cdot99}+ \\&&\gray{3\cdot9}+ \\&& {4}+{1}+{5}+{4}+{3}+{1} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${415431}$ is divisible by $9$ if ${ 4}+{1}+{5}+{4}+{3}+{1}$ is divisible by $9$ Add the digits of ${415431}$ $ {4}+{1}+{5}+{4}+{3}+{1} = {18} $ If ${18}$ is divisible by $9$ , then ${415431}$ must also be divisible by $9$ ${18}$ is divisible by $9$, therefore ${415431}$ must also be divisible by $9$.